Spectral clustering is based on algebraic graph theory. It turns the clustering problem into the graph partitioning problem. To solve the graph cut objective function, the properties of the Rayleigh quotient are usually utilized to map the original data points into a lower dimensional eigen-space by calculating the eigenvectors of Laplacian matrix and then conducting the clustering in the new space. However, during the process of spectral clustering, the space complexity of storing similarity matrix is O(n²), and the time complexity of the eigen-decomposition of Laplacian matrix is usually O(n³). Such complexity is unacceptable when dealing with large-scale data sets. It can be proved that both normalized cut graph clustering and weighted kernel k-means are equivalent to the matrix trace maximization problem, which suggests that weighted kernel k-means algorithm can be used to optimize the objective function of normalized cut without the eigen-decomposition of Laplacian matrix. Nonetheless, weighted kernel k-means algorithm needs to calculate the kernel matrix, and its space complexity is still O(n²). To address this challenge, this study proposes an approximate weighted kernel k-means algorithm in which only part of the kernel matrix is used to solve big data spectral clustering problem. Theoretical analysis and experimental comparison show that approximate weighted kernel k-means has similar clustering performance with weighted kernel k-means algorithm, but its time and space complexity is greatly reduced.